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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Grothendieck–Katz p-curvature conjecture ： ウィキペディア英語版 Grothendieck–Katz p-curvature conjecture In mathematics, the Grothendieck–Katz p-curvature conjecture is a local-global principle for linear ordinary differential equations, related to differential Galois theory and in a loose sense analogous to the result in the Chebotarev density theorem considered as the polynomial case. It is a conjecture of Alexander Grothendieck from the late 1960s, and apparently not published by him in any form. The general case remains unsolved, despite recent progress; it has been linked to geometric investigations involving algebraic foliations. ==Formulation== In a simplest possible statement, in which the p-curvature is not explicit, it can be stated in its essentials for a vector system written as :$dv/dz\; =\; A(z)v\backslash$ for a vector ''v'' of size ''n'', and an ''n''×''n'' matrix ''A'' of algebraic functions with algebraic number coefficients. The question is to give a criterion for when there is a ''full set'' of algebraic function solutions, meaning a fundamental matrix (i.e. ''n'' vector solutions put into a block matrix). For example, a classical question was for the hypergeometric equation: when does it have a pair of algebraic solutions, in terms of its parameters? The answer is known classically as Schwarz's list. In monodromy terms, the question is of identifying the cases of finite monodromy group. By reformulation and passing to a larger system, the essential case is for rational functions in ''A'' and rational number coefficients. Then a necessary condition is that for almost all prime numbers ''p'', the system defined by reduction modulo ''p'' should also have a full set of algebraic solutions, over the finite field with ''p'' elements. Grothendieck's conjecture is that these necessary conditions, for almost all ''p'', should be sufficient. The connection with ''p''-curvature is that the mod ''p'' condition stated is the same as saying the ''p''-curvature, formed by a recurrence operation on ''A'',〔(Daniel Bertrand, Bourbaki Seminar 750, 1991-2 ), section 5.〕 is zero; so another way to say it is that ''p''-curvature of 0 for almost all ''p'' implies enough algebraic solutions of the original equation. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Grothendieck–Katz p-curvature conjecture」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.